The Symplectic Structure of Curves in Three Dimensional Spaces of Constant Curvature and the Equations of Mathematical Physics

www.elsevier.com/locate/anihpc

The symplectic structure of curves in three dimensional spaces of constant curvature and the equations of mathematical physics

V. Jurdjevic

University of Toronto, Toronto, Ontario, Canada

Received 6 November 2007; received in revised form 22 November 2008; accepted 27 December 2008 Available online 7 February 2009

Abstract

The paper defines a symplectic form on an infinite dimensional Fréchet manifold of framed curves over the three dimensional space forms. The curves over which the symplectic form is defined are called horizontal-Darboux curves. It is then shown that the projection on the Lie algebra of the Hamiltonian vector field associated with the functionalf =¹_2L

0 κ²(s) dssatisfies Heisen- berg’s magnetic equation (HME), ^∂Λ_∂t (s, t)= ¹_i[Λ(s),^∂_∂s^2Λ2(s, t)]in the space of Hermitian matrices for the hyperbolic and the Euclidean case, and ^∂Λ_∂t(s, t)= [Λ(s),^∂_∂s^2Λ₂(s, t)]in the space of skew-Hermitian matrices for the spherical case. It is then shown that the horizontal-Darboux curves are parametrized by curves inSU₂, which along the solutions of (HME) satisfy Schroedinger’s non-linear equation (NSL)

−i∂ψ

∂t (t, s)=∂²ψ

∂s²(t, s)+1 2

|ψ (t, s)|²+c ψ (t, s) It is also shown that the critical points of¹_2L

0 κ²(s) ds, known as the elastic curves, correspond to the soliton solutions of (NSL).

Finally the paper shows that the modifed Korteweg–de Vries equation or the curve shortening equation are Hamiltonian equations generated byf₁=_L

0 κ²(s)τ (s) dsandf₂=_L

0 τ (s) dsand thatf₀=¹_2L

0 κ²(s) ds,f₁andf₂are all in involution with each other.

©2009 Elsevier Masson SAS. All rights reserved.

Keywords:Lie groups; Lie algebras; Symmetric spaces; Orthonormal frame bundles; Fréchet spaces; Symplectic forms; Hamiltonian vector fields

1. Introduction

This paper defines a symplectic form on an infinite dimensional Fréchet manifold of framed curves of fixed length over a three dimensional simply connected Riemannian manifold of constant curvature. The framed curves are an- chored at the initial point and are further constrained by the condition that the tangent vector of the projected curve coincides with the first leg of the orthonormal frame. Such class of curves are called anchored Darboux curves and in particular include the Serret-Frenet framed curves.

E-mail address:jurdj@math.toronto.edu.

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.12.006

The symplectic formω is defined on the space of “horizontal” curves of fixed length in the universal covers of the orthonormal frame bundles of the underlying manifolds:SL2(C)for the hyperboloidH³andSU2×SU2for the sphereS³. The formωis left invariant and is induced by the Poisson–Lie bracket on the appropriate Lie algebra. More precisely, the formωin each of the above cases is defined over the curves whose tangents take values in the Cartan spacepcorresponding to the decomposition

g=p+k

of the Lie algebragsubject to the usual Lie algebraic relations [p,p] =k, [p,k] =p, [k,k] =k.

In the case of the hyperboloid gis equal tosl2(C) and the Cartan spacep is equal to the space of the Hermitian matrices, while in the case of the sphere gis equal to su2×su2 and the Cartan space is isomorphic to the space of skew-Hermitian matrices h. The symplectic forms in each of these two cases are isomorphic to each other as a consequence of the isomorphism betweenpandhgiven byih=p.

The Euclidean spaceE³ is identified withpequipped with the metric defined by the trace form, and its framed curves are represented in the semidirect product pSU2. The Euclidean Darboux curves inherit the hyperbolic symplectic formωwhich is isomorphic to the symplectic form used by J. Millson and B. Zombro in [16].

Each group Gmentioned above is a principal SU2-bundle over the underlying symmetric space with a natural connection defined by the left invariant vector fields that take values in the Cartan spacep. The vertical distribution is defined by the left invariant vector fields that take their values ink. In this setting then, anchored Darboux curves are the solutions inGof a differential equation

dg

ds(s)=g(s)

E₁+u₁(s)A₁+u(s)A₂+u₃(s)A₃

(1) with g(0)=I, whereE₁ is a fixed unit vector in the Cartan space p. The matrices A₁, A₂, A₃ denote the skew- Hermitian Pauli matrices, andu₁(s), u₂(s), u₃(s)are arbitrary real valued functions on a fixed interval[0, L]. Each anchored Darboux curve defines a horizontal-Darboux curveh(s)∈Gthat is a solution of the differential equation

dh

ds(s)=h(s)Λ(s), Λ(s)=R(s)E1R⁻¹(s) (2)

withR(s)the solution curve inSU2of the equation dR

ds =R(s)

u₁(s)A₁+u₂(s)A₂+u₃(s)A₃

(3) that satisfiesR(0)=I. The symplectic form for the hyperbolic Darboux curves is given by

ω_Λ(V₁, V₂)=1 i L

0

Λ(s),

U₁(s), U₂(s) ds (4)

withU₁(s)andU₂(s)Hermitian matrices orthogonal to the tangent vectorΛ(s), that further satisfyU_j(0)=0 and

dV_j

ds (s)=U_j(s)forj =1,2.

In the spherical case the symplectic form has the same form as in the hyperbolic case, except for the factor ¹_i, which is omitted. The matricesU_jin this case take values inkand satisfy

dV_j ds (s)=

Λ(s), V_j(s)

+U_j(s) forj=1,2.

The second part of the paper is devoted to the Hamiltonian flow associated with the function f

g(s)

=1 2 L

0

dΛ ds (s)

²ds=1 2

L

0

κ²(s) ds

wheregdenotes a frame-periodic horizontal-Darboux periodic curve, i.e., a Darboux curve for which the solution R(s)of Eq. (3) is periodic. Hereκ(s)denotes the curvature of the projected curvex(s)in the underlying symmetric space.

It is shown that the Hamiltonian flow induced by the symplectic formωgenerates Heisenberg’s magnetic equation in the Cartan spacepgiven by

∂Λ

∂t (s, t )=1 i

Λ(s),∂²Λ

∂s²(s, t )

(5) in the hyperbolic and the Euclidean case, and by

∂Λ

∂t (s, t )=

Λ(s),∂²Λ

∂s²(s, t )

in the spherical case.

It is also shown that the corresponding matrixR(s, t )defines a complex function ψ (s, t )=u(s, t )exp

i

s

0

u₁(x, t ) dx

(6) withu(s, t )=u₂(s, t )+iu₃(s, t )that is a solution of the non-linear Schroedinger’s equation

−i∂ψ

∂t (t, s)=∂²ψ

∂s²(t, s)+1

2ψ (t, s)²+c(t )

ψ (t, s) (7)

wherec(t )= −|u(0, t )|²(Theorem 5).

This finding clarifies a remarkable observation of H. Hasimoto [7] that the function ψ (s, t )=κ(s, t )exp

i

s

0

τ (x, t ) dx

whereκ(t, s)andτ (t, s)are the curvature and the torsion of a curveγ (t, s)that evolves according to the filament equation

∂γ

∂t(t, s)=κ(t, s)B(t, s) (8)

is a solution of the non-linear Schroedinger equation (7). Indeed, when the frameR(s)in Eq. (3) is a Serret–Frenet frame thenψgiven by (6) coincides with Hasimoto’s function in the hyperbolic and the Euclidean case but not in the spherical case sinceu₁(t )=τ+¹₂.

The curves that correspond to the critical points of f = ¹_2L

0 κ²(s) ds are called elastic. The material in Sec- tion 5 shows that the elastic curves with periodic curvatures always generate soliton solutions for the non-linear Schroedinger’s equation. The elastic curves that generate solitons reside on a fixed energy level and propagate with the speed equal toH₁, whereH₁is a conserved quantity for the elastic problem. The fact that the equations for the heavy top form an invariant subsystem of the equations for the elastic curves makes the connection between elastic curves and solitons even more intriguing: the speed of the soliton corresponds to the angular momentum along the axis of symmetry for the top of Lagrange.

The formalism of this paper suggests that there is a class of functionsf₀, f₁, f₂, . . .over the space of Darboux curves that begins withf₀=¹_2L

0 κ²(s) dshaving the property that any two functions Poisson commute. It is shown in the paper thatf1andf2given by

f₁=i L

0

Λ(s),dΛ ds (s)

,d²Λ

ds²(s)

ds, f₂= L

0

d²Λ dt²

²−5 4

dΛ dt

⁴ ds

are in this class.

The above functions can be expressed either in terms of the geometric invariants of the underlying Darboux curve as:

f₁= L

0

κ²(s)τ (s) ds, f₂= L

0

∂κ

∂s

2

(s)+κ²(s)τ²(s)−1 4κ⁴(s)

ds

in which case they agree with the first three functions on the list presented by J. Langer and R. Perline in [14], or they can be expressed in terms of the complex functionu(s)defined by Eq. (6) asf₀=¹_2L

0 |u(s)|²dsand f₁= 1

2i L

0

(u¯u˙−uu) ds,˙¯ f₂= L

0

∂u

∂s(s, t ) ²−1

4u(s, t )⁴ ds

in which case they correspond to the first three conserved quantities, the number of particles, the momentum and the energy, in the paper by C. Shabat and V. Zakharov in [17].

The paper is organized as follows. Section 2 consists of geometric preliminaries leading up to the basic principal bundles in terms of which horizontal-Darboux curves are defined. Section 3 describes the symplectic structure for the space of horizontal-Darboux curves.

Section 4 is devoted to the Hamiltonian flow corresponding to f0=¹_2L

0 κ²(s) ds. This section also contains a discussion of the Euclidean symplectic form and its connection to the existing results in the literature. Section 5 deals with elastic curves and the soliton solutions for the non-linear Schroedinger’s equation. The final section (Section 6) contains a brief discussion of the conservation laws associated with f0 and their connections to the hierarchies of functions presented in [17] and [14].

2. Darboux curves and their symplectic forms 2.1. Notations and geometric preliminaries

For the purposes of this paper it will be most convenient to realize the three dimensional sphereS³and the three dimensional hyperboloidH³as subsets ofSL2(C)via the identification of pointsz=(z₀, z₁, z₂, z₃)inC⁴with the matricesZinSL2(C)through

Z=

z₀+iz₁ z₂+iz₃

−z₂+iz₃ z₀−iz₁

, z²₀+z²₁+z²₂+z²₃=Det(Z)=1.

ThenS³= {x∈R⁴: x₀²+x₁²+x₂²+x₃²=1}is identified with matricesX= _{u v}

−¯vu¯

inSU2whenzis restricted to R⁴while the hyperboloidH³= {x∈R⁴: x₀²−x₁²−x₂²−x₃²=1, x0>0}is identified with positive definite Hermitian matrices

P =

x₀+x₁ x₂+ix₃ x₂−ix₃ x₀−x₁

, Det(P )=1, by settingz₀=x₀, z₁= −ix₁, z₂=ix₃, z₃= −ix₂.

For notational convenienceSL2(C)will be denoted byGand its Lie algebra byg. Thengis the direct sum of the space of Hermitian matricespof trace zero and the subalgebra of skew-Hermitian matricesh, the Lie algebra ofSU2. The following relations are basic:

[p,p] =h, [p,h] =p, [h,h] =h, ip=h. (9) Matrices

B₁=1 2

1 0 0 −1

, B₂=1 2

0 −i

i 0

, B₃=1 2

0 1 1 0

,

known as the Hermitian Pauli matrices, form a basis forpwhile the skew-Hermitian Pauli matrices A₁=iB₁, A₂=iB₂, A₃=iB₃

form a basis forh. Together these matrices form a basis forgand conform to the following Lie bracket table:

Table 1

[ , ] A₁ A₂ A₃ B₁ B₂ B₃

A₁ 0 −A₃ A₂ 0 −B₃ B₂

A₂ A₃ 0 −A₁ B₃ 0 −B₁

A₃ −A₂ A₁ 0 −B₂ B₁ 0

B₁ 0 −B₃ B₂ 0 A₃ −A₂

B₂ B₃ 0 −B₁ −A₃ 0 A₁

B₃ −B₂ B₁ 0 A₂ −A₁ 0

Nota bene. In this paper the Lie bracket is defined as[A, B] =BA−AB.

Definition 2.1.The quadratic form ongdefined byA, B =2 Trace(AB)will be called the trace form.

The trace form is invariant in the sense that A,[B, C] =

[A, B], C, and

gAg^∗, gBg^∗ = A, B (10)

for any matricesA, B, Cing, and anyginSU2. It follows that

A, B =a₁b₁+a₂b₂+a₃b₃ (11)

for any Hermitian matricesA=₃

i=1a_iB_i andB=₃

i=1b_iB_i. SinceiA, iB = −A, Bsimilar formula holds on hwith the sign reversed.

Definition 2.2.The restriction of the trace form topwill be denoted by,hand,s will denote the negative of the restriction of the trace form toh. Then hand swill denote the induced norms onpandh.

It follows that Hermitian Pauli matrices form an orthonormal basis forprelative to,hand that skew-Hermitian Pauli matrices form an orthonormal basis onhrelative to,s.

Throughout the paperg^∗ denotes the Hermitian transpose of a matrixg. Theng∈SU2wheneverg^∗=g⁻¹. We now pass to the universal covers SU2×SU2 and SL2(C) of the orthonormal frame bundles of the sphere or the hyperboloid.

These groups will be considered principalSU2-bundles overS³andH³respectively via the following construc- tions.

ForG=SU2×SU2the action ofSU2isR(p, q)=(pR^∗, qR^∗)for each(p, q)inSU2×SU2and eachR∈SU2, and the projection mapπ is given byX=π(p, q)=pq^∗.

ForG=SL2(C)the action is(R, g)→gR^∗ for allg∈GandR∈SU2, and the projection mapπ is given by π(g)=gg^∗. In the material below we will rely on the notion of a connection on a principal bundle (in that context see [18]).

Definition 2.3.Curvesg(t )=(p(t ), q(t ))inSU2×SU2will be called spherical horizontal if p^∗dp

dt(t )=P (t ), q(t )^∗q

dt(t )= −P (t )

for some curveP (t )inh.The left invariant distributionHs((p, q))= {(pP , q(−P )): P ∈h}inSU2×SU2will be called the spherical connection.

Definition 2.4.Curvesg(t )inSL2(C)will be called hyperbolic horizontal if g⁻¹(t )dg

dt(t )=B(t )

for some curve of matricesB(t )inp. The left invariant distributionHh(g)= {gB: B∈p}will be called the hyperbolic connection.

Definition 2.5.

(a) The length of any spherical horizontal curve(g₁(t ), g₂(t ))in an interval[0, T]is equal to_T

0 P (t )sdt.

(b) The length of a hyperbolic horizontal curveg(t )in[0, T]is equal to_T

0 B(t )hdt.

It can be easily shown that the projection X(t )= _{x0+ix1 x2+ix3}

−x2+ix3x0−ix1

on S³ of any spherical horizontal curve (p(t ), q(t ))is a solution of ^dX_dt(t )=X(t )(2q(t )P (t )q(t )^∗)and

T

0

dx₀

ds

2

+dx₁ ds

2

+dx₂ ds

2

+dx₃ ds

2

dt= T

0

P (t )

sdt.

Similarly the length of a hyperbolic horizontal curve coincides with the Riemannian length T

0

−dx₀ ds

2

+dx₁ ds

2

+dx₂ ds

2

+dx₃ ds

2

dt

of the projected curveX(t )=g(t )g^∗(t ).

It can also be shown that every curve in the base manifold can be lifted to a horizontal curve and that any two liftings differ by an element inSU2, consistent with the general theory of principal bundles.

Remark 1.The preceding paragraphs reveal that the metric induced by trace form differs by a factor of 2 from the natural metric on the base manifold inherited from either the Euclidean or the Lorentzian metric inR⁴. The present choice of the metric offers some conveniences on the level of Lie algebras that the other choice does not do. For instance, Pauli matricesA1, A2, A3form an orthonormal basis relative to the trace metric and the coordinates of the matrices inhrelative to the Pauli matrices satisfy the property that the Lie bracket coincides with the cross product—a fact that is important for this paper. Relative to the natural metric on the sphere, however, vectorsE_i=2Ai,i=1,2,3, are orthonormal but then[E_i, E_j] = −2Ek and the correspondence with the cross product is changed.

2.2. Darboux curves

It follows from above that the Riemannian metric of the base manifold is induced by the left invariant metric defined on the connection distributions in terms of the trace form. On the sphere each pair(p, q)inSU2×SU2defines an orthonormal frame(v₁, v₂, v₃)atX=pq^∗where

v₁=2pA1q^∗=2pq^∗ qA₁q^∗

, v₂=2pA2q^∗=2q^∗ qA₂q^∗

, v₃=2pA3q^∗=2pq^∗ qA₃q^∗

. Conversely, every orthonormal frame at a point X ∈SU2 can be represented by the tangent vectors v1=2XU1, v2=2XU2,v3=2XU3for some matricesU1, U2, U3inhthat are orthonormal relative to the trace form. There are exactly two matrices±q∈SU2such that

U1=qA1q^∗, U2=qA2q^∗, U3=qA3q^∗. Having foundq,pis uniquely defined byp=Xq.

In the case of the hyperboloidSO(1,3)is the orthonormal frame bundle ofH³andSL2(C)is its double cover. We will identify eachginSL2(C)with the frame

v₁=2gB1g^∗, v₂=2gB2g^∗, v₃=2gB3g^∗ (12)

at X=gg^∗. Conversely every orthonormal framev₁, v₂, v₃ at a point X∈H³ can be identified with exactly two matrices±g∈SL2(C)via the above relations.

Definition 2.6.Curvesg(t ) in the universal covers of the orthonormal frame bundle will be called framed curves.

Framed curves defined on a fixed interval [0, L] which define an orthonormal frame v₁, v₂, v₃ at the base curve X(s)in the underlying symmetric space such that^dX_ds =v₁(s)will be called Darboux. Darboux curves which satisfy g(0)=I will be called anchored.

Remark 2. Condition ^dX_ds =v₁(s)implies that X(s) is parametrized by arc length and thereforeL is the length ofX(s). The fact that the orthonormal bundles are replaced by their universal covers does not matter in the subsequent exposition since all Darboux curves will be anchored.

In the spherical case anchored Darboux curvesg(s)=(p(s), q(s))are the solutions of dp

ds(s)=p(s)P (s), dq

ds(s)=q(s)Q(s), P (s)−Q(s)=2A1

satisfying the initial conditionsp(0)=I,q(0)=I. ConditionP (s)−Q(s)=A1can be expressed also as P (s)=U (s)+A₁, Q(s)=U (s)−A₁, whereU (s)=1

2

P (s)+Q(s)

. (13)

Definition 2.7.Anchored spherical Darboux curves are said to be reduced if the curve U (s)in (13) is of the form U (s)= _{0 u(s)}

− ¯u(s) 0

for some complex curveu(s).

Every anchored Darboux curve(p(s), q(s))can be transformed into a reduced Darboux curve(p(s),˜ q(s)), without˜ altering the base curveX(s), by takingp˜=ph,q˜=qhwithh(s)the solution of^dh_ds = −h(s)_{iu1(s) 0}

0 −iu1(s)

, h(0)=I where the matrix_{iu1(s) 0}

0 −iu1(s)

denotes the diagonal part ofU (s). ThusX(s)can be lifted also to a reduced Darboux curve. On the other hand, reduced Darboux framed curves exclude the Serret–Frenet frames as we will see later on.

The significance of these observations will become clear further on in the paper.

Definition 2.8.Curves(p(s), q(s))inSU2×SU2which are the solutions of d

ds

p(s), q(s)

=

p(s), q(s)

Λ(s),−Λ(s)

, Λ(0)=A₁, Λ(s)=1, p(0)=q(0)=I (14) will be called spherical horizontal-Darboux curves.

Every anchored spherical Darboux curve g(s)=(p(s), q(s)) can be transformed into a spherical horizontal- Darboux curve

˜

p(s)=p(s)R^∗(s), q(s)˜ =q(s)R^∗(s)

for some matrixR(s)∈SU2, R(0)=I without altering the projected curveX(s)=p(s)q^∗(s). In fact, R(s)is a solution of ^dR_ds =¹₂R(s)(P (s)+Q(s)), andp˜andq˜are the solutions of

dp˜ ds = ˜p

1

2R(P −Q)R^∗

= ˜p

RA1R^∗

, dq˜ ds = ˜q

1

2R(Q−P )R^∗

= ˜q

−RA1R^∗

. (15)

Conversely, every curveΛ(s)∈hwithΛ(s) =1, Λ(0)=A₁can be written asΛ(s)=R(s)A₁R^∗(s)for some curve R(s)inSU2withR(0)=I becauseSU2acts transitively by conjugations on the sphereΛ =1. The correspondence betweenΛandRis not bijective: ifR₀→ΛthenR₀h→Λfor anyh=_z0

0z¯

,z =1.

Curves R(s)defined by Λ(s)=R(s)A₁R^∗(s)withR(0)=I define spherical Darboux curves (p(s), q(s))via the relations (13) where U (s)=R^∗(s)^dR_ds(s). If the diagonal part of U (s) is equal to zero then (p(s), q(s)) is a reduced Darboux curve. It follows that such curves set up a bijective correspondence between the horizontal-Darboux curves and the reduced Darboux curves. Thus every curveX(s)parametrized by arc length on the interval[0, L]with boundary conditionsX(0)=I and ^dX_ds(0)=2A1can be lifted to a unique spherical horizontal-Darboux curve and also to a unique reduced anchored spherical Darboux curve.

In the subsequent exposition we will be less formal and refer to the spherical horizontal-Darboux curves as the solutions of the initial value problem

dp

ds(s)=p(s)Λ(s), Λ(s)=1, p(0)=I, (16)

since then the second factorq(s)is defined by of^dq_ds =q(s)(−Λ(s)),q(0)=I.

Definition 2.9.Spherical horizontal-Darboux curvesp(s)for whichΛ(s)=R(s)A₁R^∗(s)for some curveR(s)∈SU2

such thatR(L)=R(0)=I are called frame-periodic.

Remark 3. Frame-periodicity implies not only thatΛ(s)is periodic but also implies that the corresponding Dar- boux curve (p(s), q(s)) is a solution of an equation with periodic right-hand side since the matrix U (s) is peri- odic. However, if U (s) is periodic its diagonal partD=_{iu1 0}

0 −iu₁

is periodic and therefore h(s), the solution of

dh

ds = −h(s)D(s), h(0)=Isatisfiesh(L)=I, from which it follows that the reduced Darboux curve that corresponds to the horizontal-Darboux curve has periodic right-hand side as well.

Definition 2.10. The set of anchored spherical Darboux curves will be denoted by Ds(L). The set of spherical horizontal-Darboux curves will be denoted byHDs(L)and the set of frame-periodic horizontal curves byPHDs(L).

In the case of the hyperboloid an anchored Darboux curveg(s)∈SL2(C)defines framesv₁(s)=2g(s)B1g^∗(s), v₂=2g(s)B2g^∗(s),v₃(s)=2g(s)B3g^∗(s)over the projected curveX(s)=g(s)g^∗(s)such that ^dX_ds =2g(s)B1g^∗(s).

It then follows that dg

ds =g(s)

B1+A(s)

(17) for some matrix curveA(s)inhfor the following reasons:

If ^dg_ds =g(s)(B(s)+A(s))withB(s)∈pandA(s)∈h, and ifg(s)˜ =g(s)R⁻¹(s)for someR(s)∈SU2then both gandg˜project onto the same curveX(s). In particular if^dR_ds =R(s)A(s)then ^d_ds^g˜= ˜g(s)(R(s)B(s)R^∗(s)).Hence

dX

ds =2g(s)˜

R(s)B(s)R^∗(s)

˜

g^∗(s)=2g(s)B(s)g^∗(s)=2g(s)B1g^∗(s), and thereforeB(s)=B₁.

Similar to the spherical case, hyperbolic Darboux curves for which the diagonal part of the matrixAis equal to zero will be called reduced. It follows that any base curve X(s)of an anchored Darboux curve is initially fixed at X(0)=I and has a fixed initial tangent vector ^dX_ds(0)=2B1. Furthermore, it follows from above thatX(s) is the projection of a horizontal curveg(s)˜ such that

˜ g^∗dg˜

ds(s)=Λ(s)=R(s)B₁R^∗(s) for some curveR(s)inSU2.

Definition 2.11.Hyperbolic horizontal curvesg(s)will be called hyperbolic horizontal-Darboux ifg(0)=I and g⁻¹(s)dg

ds(s)=Λ(s), Λ(s)∈p, Λ(s)

h=1, Λ(0)=B₁. (18)

It follows that every curveX(s)on the hyperboloid parametrized by arc length on[0, L]that satisfiesX(0)=I and ^dX_ds(0)=2B1 is the projection of a unique hyperbolic horizontal-Darboux curve g(s). Moreover, the relation Λ(s)=R(s)B₁R^∗(s),R(0)=I defines an anchored hyperbolic curveg˜=gRoverX. As in the spherical case, the correspondence between hyperbolic horizontal-Darboux curves and reduced hyperbolic anchored Darboux curves is bijective.

It follows from above that the horizontal-Darboux curves in both the spherical and the hyperbolic case are parametrized by matrices R(s)inSU2which are solutions of ^dR_ds =R(s)U (s),R(0)=I, withU (s)= _{0 u(s)}

− ¯u(s) 0

for some complex curveu(s)through the relations

dg

ds =Λ(s)=R(s)CR^∗(s), R(0)=I (19)

whereC=A₁in the spherical case andC=B₁in the hyperbolic case.

Definition 2.12. Hyperbolic horizontal-Darboux curvesg(s) are frame-periodic ifΛ(s)in (2.11) satisfiesΛ(s)= R(s)B₁R^∗(s)for some curveR(s)∈hsuch thatR(0)=R(L)=I.

Definition 2.13.The space of all anchored hyperbolic Darboux curves will be denoted byDh(L), the space of hy- perbolic horizontal-Darboux, respectively frame-periodic hyperbolic horizontal-Darboux curves will be denoted by HDh(L)andPHDh(L).

In both the spherical and the hyperbolic case frame-periodicity implies that the matrixU (s)=R(s)^∗dR_ds is smoothly periodic. The same applies to the matrixΛ(s)=R(s)A₁R^∗(s)(respectivelyΛ(s)=R(s)B₁R^∗(s)). This implies that the projections of frame-periodic curves necessarily have periodic curvature and torsion, but need not be closed.

Conversely, all smoothly periodic curves have periodic curvature and torsion. However, it might not be true that smooth periodic curves in the base space lift to frame periodic curves in the orthonormal frame bundle.

3. Darboux curves as Fréchet manifolds

On the basis of the general theory developed in [6] each space of anchored or frame-periodic Darboux curves and their horizontal projections can be considered as an infinite-dimensional Fréchet manifold. Recall that a topological Hausdorff vector spaceV is called a Fréchet space if its topology is induced by a countable family of semi-normsp_n, and if it is complete relative to the semi-norms in{pn}. A Fréchet manifold is defined as follows:

Definition 3.1.A Fréchet manifold is a topological Hausdorff space equipped with an atlas whose charts take values in open subsets of a Fréchet spaceV such that any change of coordinate charts is smooth.

The paper of R.S. Hamilton [6] singles out an important class of Fréchet manifolds, called tame, in which the implicit function theorem is true. One of the main theorems in [6] is that the set of smooth mappings from a compact interval into a finite-dimensional Riemannian manifoldMis a tame Fréchet manifold. It therefore follows from the implicit function theorem that closed subsets of tame Fréchet manifoldsM, defined by the zero sets of finitely many smooth functions onMare tame sub-manifolds ofM. Since the anchored Darboux curves are particular cases of the above situation, it follows that each of them is a tame Fréchet manifold and the same applies to their horizontal projections. Tangent vectors and tangent bundles of Fréchet manifolds are defined in the same manner as for finite dimensional manifolds. In particular tangent vectors at a pointxin a Fréchet manifoldMare the equivalence classes of curves σ (t )inMall emanating from x (i.e.,σ (0)=x), and all having the same tangent vector ^dσ_dt(0)in each equivalence class. The set of all tangent vectors atxdenoted byT_xMconstitutes the tangent space atx.

The tangent bundle of a Fréchet manifoldMis a Fréchet manifold. A vector fieldXonMis a smooth mapping fromMinto the tangent bundleTMsuch thatX(x)∈T_xMfor eachx∈M. On tame Fréchet manifolds vector fields can be defined as derivations in the space of smooth functions onM.

3.0.1. Tangent spaces for horizontal-Darboux curves

The calculations in this section make use of covariant derivatives which are recalled below for reader’s convenience.

Definition 3.2.

(a) The covariant derivative of a curve of tangent vectorsv(s)=X(s)U (s)along a curveX(s)inSU2is given by D_X

ds (v)(s)=X(s) dU

ds +1 2

U (s), Λ(s)

(20) whereΛ(s)=X^∗(s)^dX_ds(s).

(b) The covariant derivative^D_ds^g(v)of a curve of tangent vectorsv(s)=g(s)U (s),U (s)∈p, along a horizontal curve g(s)inSL2(C), is defined by

D_g

ds(v)(s)=g(s)dU

ds(s) (21)

for alls∈ [0, L].

The reader can easily verify that the covariant derivative on SU2 is equal to the orthogonal projection of the ordinary derivative in R⁴ onto the tangent space of the sphere when the sphere is considered a submanifold ofR⁴. On the hyperboloid, however, the notion of covariant derivative for vectors in the horizontal distributionHhcoincides with the usual notion of covariant derivative in the base manifoldH³in the sense that

D_π(g) ds

π_∗(gV )

(s)=π_∗

g(s)dV ds

(s).

The subsequent material also makes use of the following

Lemma 1.Suppose thatX(s, t )is a field of curves in SU2with its infinitesimal directions

A(s, t )=X^∗(s, t )∂X

∂s(s, t ) and B(s, t )=X^∗(s, t )∂X

∂t (s, t ).

Then

∂A

∂t −∂B

∂s + [A, B] =0. (22)

Proof. On any Riemannian manifold ^D_ds^X(^∂X_∂t )=^D_ds^X(^∂X_∂s).Hence, X

∂B

∂s +1 2[B, A]

=X ∂A

∂t +1 2[A, B]

and therefore,

∂A

∂t −∂B

∂s + [A, B] =0. 2

Eq. (22) is also known as the zero-curvature equation [5].

Theorem 1.

(a) The tangent spaceT_p(HDs)(L)at a spherical horizontal-Darboux curvep(s)withΛ(s)=p^∗(s)^dp_ds(s)consists of curvesv(s)=X(s)V (s)withV (s)the solution of

dV ds (s)=

Λ(s), V (s)

+U (s) (23)

such thatV (0)=0, whereU (s)is a curve inhsubject to the conditions thatU (0)=0andΛ(s), U (s)s=0.

(b) Tangent vectorsv(s)=X(s)V (s)at frame-periodic horizontal-Darboux curvesX(s)are generated by smoothly periodic curvesU (s)whose period is equal toL.

Proof. LetY (s, t )denote a family of anchored horizontal-Darboux curves such thatY (s,0)=p(s). Then, v(s)=

∂Y

∂t(s, t )_t=0is a tangent vector atX(s)for whichv(0)=0 since the curvesY (s, t )are anchored.

LetZ(s, t )andW (s, t )denote the matrices defined by Z(s, t )=Y (s, t )^∗∂Y

∂s(s, t ), W (s, t )=Y (s, t )^∗∂Y

∂t(s, t ).

It follows thatΛ(s)=Z(s,0),V (s)=W (s,0). Equation

∂Z

∂t −∂W

∂s + [Z, W] =0 fort=0 reduces to

dV ds(s)=

Λ(s), V (s) +U (s) whereU (s)=^∂Z_∂t(s,0).

Since the curvess→Y (s, t )are Darboux for eacht,Z(s, t ), Z(s, t )s=1 andZ(0, t )=A₁. Therefore,

Z(s, t ),∂Z

∂t (s, t )

=0, and ∂Z

∂t (0, t )=0 which implies thatΛ(s), U (s)s=0 andU (0)=0.

It remains to show that any curveV (s)inhthat satisfies (23) can be realized by the perturbationsY (s, t )used above.

So assume thatV (s)be any solution of (23) generated by a curveU (s)withU (0)=0 that satisfiesΛ(s), U (s)s=0.

Letφ(t)denote any smooth function such thatφ(0)=0 and ^dφ_dt(0)=1. Define

Z(s, t )= 1

1+φ²(t )U (s), U (s)s

Λ(s)+φ(t)U (s) .

Evidently Z(0, t )=A1 for allt, and an easy calculation shows thatZ(s, t ), Z(s, t )s =1. ThereforeY (s, t ), the solution of

∂Y

∂s(s, t )=Y (s, t )Z(s, t )

withY (0, t )=I corresponds to an anchored horizontal-Darboux curve for eacht. SinceU (s)=^∂Z_∂t(s,0)our proof of part (a) is finished.

To prove part (b) assume that the curvesY (s, t )used above belong toPHDs(L). Then,s→Z(s, t )areL-periodic for eacht, and therefore,U (s)=^∂Z_∂t(s,0)is alsoL-periodic. 2

Tangent spaces ofHDh(L)andPHDh(L), obtained along similar lines as in the spherical case, are described by the following

Theorem 2.

(a) The tangent spaceT_g(HDh)(L)at a hyperbolic horizontal-Darboux curveg(s)withΛ(s)=g⁻¹(s)^dg_ds(s)consists of curvesv(s)=g(s)V (s)such that

dV

ds =U (s), V (0)=0, (24)

whereU (s)is a Hermitian curve subject to the following conditions:

U (0)=0,

Λ(s), U (s)_h=0.

(b) For frame-periodic horizontal-Darboux curves the curve ^dV_ds(s) must be smoothly periodic having the period equal toL.

Proof. Let h(s, t ) denote a family of anchored horizontal-Darboux curves such thath(s,0)=g(s). Then v(s)=

∂h

∂t(s, t )_t=0is a tangent vector atg(s)such thatv(0)=0 since the curvesh(s, t )are anchored.

LetZ(s, t )andW (s, t )denote the matrices defined by Z(s, t )=h(s, t )⁻¹∂h

∂s(s, t ), W (s, t )=h(s, t )⁻¹∂h

∂t(s, t ).

It follows thatΛ(s)=Z(s,0)andv(s)=g(s)V (s)withV (s)=W (s,0). Then

∂Z

∂t (s, t )=∂W

∂s (s, t )

is the hyperbolic analogue of the zero-curvature equation (22). Fort=0 the above equation reduces to dV

ds (s)=∂W

∂s (s,0)=U (s), whereU (s)=^∂Z_∂t(s,0). Then

Z(s, t ), Z(s, t )

h=1, and Z(0, t )=B₁ imply thatΛ(s), U (s)h=0 andU (0)=0.

Conversely any curveV (s)inhthat satisfies (24) can be realized by the perturbationsh(s, t )defined in the first part of the proof. The argument is the same as in the spherical case and will be omitted. The same applies to the proof of part (b). 2

3.1. The symplectic structure of horizontal-Darboux curves

The basic notions of symplectic geometry of infinite-dimensional Fréchet manifolds are defined through differential forms in the same manner as for the finite-dimensional situations. In particular, differential formsωof degreenare mappings

ω:X (M)× · · · × X(M)

n

→C^∞(M)

that areC^∞(M)multilinear and skew-symmetric. HereX(M)denotes the space of all smooth vector fields onM.

Definition 3.3.The exterior derivativedωof a form of degreenis a differential form of degreen+1 defined by dω(X₁, . . . , X_n+1)=

n+1

i=1

(−1)ⁱ⁺¹X_i

ω(X₁, . . . ,Xˆ_i, . . . , X_n)

−

i<j

(−1)^i+jω

[X_i, X_j], . . . ,Xˆ_i, . . . ,Xˆ_j, X_n+1

where the roof sign above an entry indicates its absence from the expression (i.e., w(Xˆ1, X2)=w(X2) and w(X1,Xˆ2)=w(X1)).

A differential formωis said to be closed if its exterior derivativedωis equal to zero.

Definition 3.4.A differential formωof degree 2 is said to be symplectic whenever it is closed and non-degenerate, in the sense that the induced form(i_Xω)(Y )=ω(X, Y )is non-zero for each non-zero vector fieldX.

The differentialdf of a smooth functionf is a form of degree 1 defined bydf (v)=_dt^df◦σ (t )|t=0for any smooth curve inMsuch thatσ (0)=x, and ^dσ_dt(0)=v.

In finite dimensional symplectic manifolds with a symplectic formωthere is a unique vector fieldX_f such that df =i_Xfω.X_f is called the Hamiltonian vector field induced byf, andf is called the Hamiltonian ofX_f. However, in infinite dimensional manifolds it may happen that the formdf is not equal toi_Xwfor anyX∈X(M). This is due to the fact that the cotangent bundle of an infinite dimensional Fréchet space is never a Fréchet manifold. Nevertheless, Definition 3.5.A vector fieldXis said to be Hamiltonian if there exists a smooth functionf such that

df (Y )=ω(X, Y )

for all vector fieldsY onM. The dependence ofXonf shall be noted explicitly byX_f.

The manifold consisting of horizontal-Darboux curves admits a natural 2-form ω which will be defined first for spherical horizontal-Darboux curves. In the process it will become clear how to adapt the results to hyperbolic horizontal-Darboux curves. Letv₁(s)=X(s)V₁(s)andv₂(s)=X(s)V₂(s)denote any tangent vectors at a horizontal- Darboux curvep(s)that is defined by^dp_ds(s)=p(s)Λ(s). According to (23) there exist unique curvesU1(s)andU2(s) such that

U_i(0)=0,

Λ(s), U_i(s)_s=0

and

U_i(s)=dV_i ds (s)−

Λ(s), V_i(s)

, i=1,2. (25)

Thenωis given by ω_Λ(V₁, V₂)= −

L

0

Λ(s),

U₁(s), U₂(s)

sds. (26)

Remark.As in finite dimensional situations the choice of sign is a matter of convention. The justification for the above choice of sign will be given later in the paper.

Theorem 3.BothHDs(L)andPHDs(L)are symplectic Fréchet manifolds relative toωdefined by(26).

The following lemmas will be useful for the proof of the theorem.

Lemma 2.

A,[B, C]

= A, CsB− A, BsC for any elementsA, B, Cinh.

We leave the proof to the reader.

Lemma 3.Suppose thatv(s)=g(s)V (s)is a tangent vector at a horizontal-Darboux curveg(s). LetU (s)be defined by(23). Then there exists a curveC(s)inksuch that

U (s)=

Λ(s), C(s) .

Proof. The mappingC→ [Λ(s), C(s)]restricted to the orthogonal complement ofΛ(s)is surjective. SinceU (s)is orthogonal toΛ(s)the proof follows. 2

Proof of the theorem. The proof is the same for each ofHDs(L)andPHDs(L) and will be presented formally without any reference to the underlying space.

Evidently ωis skew-symmetric. To show that it is non-degenerate, assume that ωΛ(V1, V )=0 for all tangent vectorsgV. LetU₁(s)correspond toV₁(s)defined by Eqs. (25). ThenU (s)= [Λ(s), U₁(s)]satisfiesU (0)=0, and Λ(s), U (s)s =0. Therefore the corresponding vectorgV withV the solution of Eq. (25) belongs to the tangent space atg. It follows from Lemma 2 that

U₁,[Λ, U₁]

= U₁, U₁sΛ= U₁²sΛ.

Therefore, Λ(s),

U₁(s), U (s)

s=Λ(s)²

sU₁(s)²

s=U₁(s)²

s

which implies thatU₁(s)=0 since 0=ω_Λ(V₁, V )=_L

0 U₁(s)²ds. But then (25) implies thatV₁(s)=0. Hence,ω is non-degenerate.

To show thatωis closed letv_i(s)=g(s)V_i(s), 1i3 denote any three tangent vectors at a fixed Darboux curve g(s). It is required to show (Definition 3.3) that

dω(X₁, X₂, X₃)=

cyclic

X_i

ω(X_j, X_k)

+

cyclic

ω

[X_i, X_j], X_k

=0 (27)

whereXi denote any vector fields such thatXi(g)=vifor eachi=1,2,3.

LetX_i(z)=zZ_i, i=1,2,3 denote vector fields over Darboux base curveszwithZ_ithe solutions of dZ_i

ds (s)=

Λ_z(s), Z_i(s)

+U_i(s) whereΛ_z=z^∗dz ds.